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Also, that tree is a great representation; it's much more clear to me now. Thanks Weregoose!
Actually, this seems like a by-far better solution. Why weren't we doing this from the beginning?
I think this requires more steps computationally, but it may be simpler to program on a computer, I don't know. Using the same inputs from earlier, make a graph like the one below, then ask, "Which Hamiltonian path gives the greatest sum?" the only restriction being that you must return to the starting point again after landing on all four vertices (which agrees with the definition of a Hamiltonian path, so rotations are automatically accounted for).



The answer should remain 01011→11110→10100→10001→01011.
That might be simple for me to program. I'll have to ruminate on the possibilities.
  
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